Abstract
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.
Highlights
Let (Sk)k 1, where Sk = X1 + . . . + Xk, be a planar random walk with independent identically distributed increments X1, X2, . . . We assume that the expectation of X1 exists and is finite, and put μ := EX1
Abstract. — We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments
For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments
Summary
We found the rate function for random walks that have rotationally invariant distributions of increments with finite Laplace transform (Theorem 2.11) and for Gaussian random walks with arbitrary drift (Theorem 2.13 and Proposition 2.15) In all these results we identified the asymptotic form of optimal trajectories of the walk resulting in the large deviations. We extended the above results on random walks, which have increments in discrete time, to convex hulls of planar Lévy processes (Theorem 2.16) with finite Laplace transform, including Brownian motions.
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